A search for single production of vector-like quarks, T and B, decaying into a Z boson and a top or a bottom quark, respectively, is presented. The search is performed using data collected by the CMS experiment at the LHC in proton-proton collisions at s=13 TeV, corresponding to an integrated luminosity of 2.3fb−1. An exotic T quark production mode through the decay of a heavy Z′ resonance is also considered. The search is performed in events with a Z boson decaying leptonically, accompanied by a bottom or a top quark decaying hadronically. No excess of events is observed over the standard model background expectation. Products of production cross section and branching fraction for T and B quarks from 1.26 and 0.13 pb are excluded at 95% confidence level for the range of resonance mass considered, which is between 0.7 and 1.7 TeV. Limits on the product of the Z′ boson production cross section and branching fraction, with the Z′ boson decaying to the Tt final state, are set between 0.31 and 0.13 pb, for Z′ boson masses in the range from 1.5 to 2.5 TeV. This is the first search at 13 TeV for single production of vector-like quarks in events with a Z boson decaying leptonically accompanied by boosted jets.[Figure not available: see fulltext.].